Native Painted Turtles (cpb) 2008-2017

Statistical Analysis of a turtle population "Chysemmys picta belli" at two locations in the pacific north west surveyed between 2008 and 2017

Data cleaning and new features

data cleaning and aggregation

Species are recorded to either Cbp (native painted turtle) or Res (red eared sliders). This analysis only looks at Cbp-Native-Turtles

Gravid True female turtle carrying eggs

Sex m, f or unknown

Carapace, Plastron, Weight cleaned decimals.. only decimals >0 utilized in analysis

Weight a value > 0 means recorded survey. Value = 0 means repeat sighting and not relevant for analisis

new features

new_annuli it is very hard to count the annuli for older surveys accurately. Different surveyors might come up with different annuli values for the same turtle. Further, it becomes harder the older the turtle gets. Thus for some old turtles, it is impossible to assess the annuli. For that reason, a new feature new_annuli is created. We conservatively take the lowest ever counted annuli for an individual turtle and then increment its annuli for each sighting in the following years. Pseudo Code:

new_annuli = (year_of_sighting - year_of_very_fist_sighting) + lowest_annuli_ever_recorded

gender_plus m, f, f_gra, unknown (distinguishes non-gravid and gravid females

import warnings
warnings.filterwarnings('ignore')

#Prepping the notebook
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
# my custom helper functions
import helpers as hlp
import turtles
import scipy
import pandas as pd
#filter data: Naitive turle and relevant
print ("Filtering Natives ...")
natives = turtles.get_clean_data()
natives = natives[natives['Weight']!=0]
natives = natives[natives['Carapace']!=0]
natives = natives[natives['Plastron']!=0]
natives = natives[natives['Species']=='Cpb']
print ("Done")

Filtering Natives ...
Loading data source/Turtle Data.xls
2008
2009
2010
2011
2012
2013
2014
Loading data source/MF Trapping Data.xlsx
Loading data source/Whitaker Trapping Data.xlsx
Concat Loaded data...
Cleaning decimals ...
Cleaning other values ...
Add new features ...
Done

EDA - Exploratory Data Analysis

basic metrics (sanity check that the data loaded correct)

#Show some data - basic metrics
natives[['Carapace','Plastron','Annuli','Weight']].describe()

	Carapace	Plastron	Annuli	Weight
count	904.000000	904.000000	904.000000	904.000000
mean	141.395432	133.651642	6.590708	479.428986
std	40.148567	39.074944	3.643979	370.193451
min	32.740002	29.730000	0.000000	7.050000
25%	111.095001	104.107502	4.000000	198.000000
50%	133.049995	125.230000	6.000000	323.000000
75%	177.507496	166.584995	9.000000	712.250000
max	237.199997	242.199997	21.000000	1435.000000

#Show some data - Gender, Location etc
natives[['Species','Capture Location','Source','ID']].pivot_table(index=['Capture Location','Source','Species'],aggfunc='count')

			ID
Capture Location	Source	Species
Gresham	Turtle Data.xls|2008	Cpb	48
	Turtle Data.xls|2009	Cpb	14
	Turtle Data.xls|2010	Cpb	48
	Turtle Data.xls|2011	Cpb	114
	Turtle Data.xls|2012	Cpb	187
	Turtle Data.xls|2013	Cpb	103
	Turtle Data.xls|2014	Cpb	225
Mason Flats	source/MF Trapping Data.xlsx	Cpb	149
Whitaker Ponds	source/Whitaker Trapping Data.xlsx	Cpb	16

Swarmplots

Swarmplots to show the distribution of Carapace, Plastron, and Weight by Gender

The following plot shows: the min/max size(weight) for each sex and how much diversity there is in weights in a given population. Most males are under 400g, females peak at 400 and 1000g.

### Basic Swarmplot to show Carapace / Plastron / Weight by Gender
plt.style.use('seaborn-notebook')
plt.rcParams['figure.dpi'] = 300
colors = [sns.xkcd_rgb['pale red'],sns.xkcd_rgb['denim blue'],sns.xkcd_rgb['medium green']]

print ("Plotting swarm plot with " + str(natives.Weight.count()) + ' samples...')
sns.set()
_ = plt.figure(1)
_ = plt.suptitle('Native Turtles Swarmplots ')

_1 =plt.subplot(2,2,1)
_ = sns.swarmplot(x='Gender', y='Carapace', data=natives,size=3,palette=colors)
_ = plt.ylabel('Carapace')

_2 =plt.subplot(2,2,2,sharey=_1)
_ = sns.swarmplot(x='Gender', y='Plastron', data=natives,size=3,palette=colors)
_ = plt.ylabel('Plastron')
_ = plt.tight_layout(rect=[0, 0.03, 1, 0.95])
_ = plt.subplot(2,2,3)
_ = sns.swarmplot(x='Gender', y='Weight', data=natives,size=3,palette=colors)
_ = plt.xlabel('Gender')
_ = plt.ylabel('Weight')

Plotting swarm plot with 904 samples...

Weight, Carapace and Plastron distibution by sex

Weight, carapace, and plastron show the same pattern - females ultimately grow larger and heavier than males. Trapping will show a greater diversity in female sizes/weights than with males. I am curious about the outliers.

#
natives_melted = natives.copy()
natives_melted['Weight / 10'] = natives_melted.Weight.div(10)
natives_melted = pd.melt(natives_melted[['Carapace','Plastron','Weight / 10','Gender']],'Gender',var_name='measurement')

_ = sns.swarmplot(x="measurement", y="value", hue="Gender", data=natives_melted,palette=colors,size=3)
_ = plt.show()

## Outliers

someColumns = [
    'Date',
    'ID',
    'Capture Location',
    'Gender',
    'date_min',
    'annuli_min',
    'Carapace',
    'Plastron',
    'Weight',
]

display(natives[
    (natives.Plastron <47) |
    (natives.Carapace <47) |
    (natives.Plastron >220) 
][someColumns])

# display(natives_melted[
#     (natives_melted.value <70) &(natives_melted.measurement != 'Weight / 10')])

	Date	ID	Capture Location	Gender	date_min	annuli_min	Carapace	Plastron	Weight
343	2010-06-10	09-09/H9	Gresham	unknown	2010-06-10	1.0	32.740002	29.730000	7.050000
602	2011-08-02	RM1, LM4	Gresham	unknown	2011-08-02	2.0	47.310001	45.500000	21.700001
1043	2017-09-29	2076	Mason Flats	f	2017-09-29	9.0	237.199997	242.199997	835.000000

ECDF Graph of size/weight distribution by sex

Shows the difference in size/weight distributions between sexes; males have a nice bell curve with an even growth rate throughout life; females have two bumps, one at 400g/150mm carapace and plastron and the other at 1000g, and 200mm shell size.

### Same information as beswars above just displayed as ECDFs
plt.style.use('seaborn-notebook')
plt.rcParams['figure.dpi'] = 300
#colors = [sns.xkcd_rgb['pale red'],sns.xkcd_rgb['denim blue']]


print ("Plotting Cumulative Distribution Function with " + str(natives.Weight.count()) + ' samples...')
# Compute ECDFs

_ = plt.figure(2)
_ = plt.suptitle('Native Turtles Cumulative Distribution ')

x_CarapaceF, y_CarapaceF = hlp.ecdf(natives[natives['Gender']=='f']['Carapace'])
x_CarapaceM, y_CarapaceM = hlp.ecdf(natives[natives['Gender']=='m']['Carapace'])

_ = plt.subplot(2,2,1)
_ = plt.plot(x_CarapaceF, y_CarapaceF, marker='.',linestyle = 'none',color=colors[0])
_ = plt.plot(x_CarapaceM, y_CarapaceM, marker='.',linestyle = 'none',color=colors[1])
_ = plt.margins(0.02)
_ = plt.legend(('Female', 'Male'), loc='lower right')
_ = plt.xlabel('Carapace (mm)')
_ = plt.ylabel('ECDF')

x_PlastronF, y_PlastronF = hlp.ecdf(natives[natives['Gender']=='f']['Plastron'])
x_PlastronM, y_PlastronM = hlp.ecdf(natives[natives['Gender']=='m']['Plastron'])

_ = plt.subplot(2,2,2)
_ = plt.plot(x_PlastronF, y_PlastronF, marker='.',linestyle = 'none',color=colors[0])
_ = plt.plot(x_PlastronM, y_PlastronM, marker='.',linestyle = 'none',color=colors[1])
_ = plt.margins(0.02)
_ = plt.legend(('Female', 'Male'), loc='lower right')
_ = plt.xlabel('Plastron (mm)')
_ = plt.ylabel('ECDF')
_ = plt.tight_layout(rect=[0, 0.03, 1, 0.95])

x_WeightF, y_WeightF = hlp.ecdf(natives[natives['Gender']=='f']['Weight'])
x_WeightM, y_WeightM = hlp.ecdf(natives[natives['Gender']=='m']['Weight'])

_ = plt.subplot(2,2,3)
_ = plt.plot(x_WeightF, y_WeightF, marker='.',linestyle = 'none',color=colors[0])
_ = plt.plot(x_WeightM, y_WeightM, marker='.',linestyle = 'none',color=colors[1])
_ = plt.margins(0.02)
_ = plt.legend(('Female', 'Male'), loc='lower right')
_ = plt.xlabel('Weight (g)')
_ = plt.ylabel('ECDF')

Plotting Cumulative Distribution Function with 904 samples...

Female size/weight vs age

It shows a similar pattern to the ECDF graphs - females show two distinct "bumps" suggesting two concentrations of female animals in the population.

#2d histogram Carapace / Plastron / Weight - Females
sns.reset_orig()
#plt.style.use('seaborn-notebook')
plt.rcParams['figure.dpi'] = 300
## Lineplot Age + Weight, Carapace, Plastron

### FEMALE
females = natives[natives['Gender']=='f']
females = females[females['Annuli']!=0]
print ("Plotting Histogram Weight, Carapace and Plastron with " + str(females.Weight.count()) + ' samples...')

_ = plt.figure(3)
_1 =plt.subplot(2,2,1)

#_ = plt.scatter(females['Annuli'], females['Carapace'])
_ = plt.hist2d(females['Annuli'], females['Carapace'],cmap='Reds',bins=(15,15))
_ = plt.colorbar()
_ = plt.ylabel('Carapace')

_2 =plt.subplot(2,2,2,sharey=_1)
#_ = plt.scatter(females['Annuli'],females['Plastron'])
_ = plt.hist2d(females['Annuli'], females['Plastron'],cmap='Reds',bins=(15,15))
_ = plt.colorbar()
_ = plt.ylabel('Plastron')

_ = plt.suptitle('Native Females Histogram ')
_ = plt.subplot(2,2,3)
#_ = plt.scatter(females['Annuli'], females['Weight'],s=10)
_ = plt.hist2d(females['Annuli'], females['Weight'],cmap='Reds',bins=(15,15))
_ = plt.colorbar()
_ = plt.xlabel('Annuli')
_ = plt.ylabel('Weight')
_ = plt.tight_layout(rect=[0, 0.03, 1, 0.95])

Plotting Histogram Weight, Carapace and Plastron with 438 samples...

Males size/weight vs age

Males show one concentration of animals, suggesting a "normal bell curve" distribution.

##2d histogram Carapace / Plastron / Weight - Males
sns.reset_orig()
#plt.style.use('seaborn-notebook')
plt.rcParams['figure.dpi'] = 300
## Lineplot Age + Weight, Carapace, Plastron

### MALE
males = natives[natives['Gender']=='m']
males = males[males['Annuli']!=0]
print ("Plotting Histogram Weight, Carapace and Plastron with " + str(males.Weight.count()) + ' samples...')

_ = plt.figure(3)

_1 =plt.subplot(2,2,1)
#_ = plt.scatter(males['Annuli'], males['Carapace'])
_ = plt.hist2d(males['Annuli'], males['Carapace'],cmap='Blues',bins=(15,15))
_ = plt.colorbar()
_ = plt.ylabel('Carapace')

_2 =plt.subplot(2,2,2,sharey=_1)
#_ = plt.scatter(males['Annuli'],males['Plastron'])
_ = plt.hist2d(males['Annuli'], males['Plastron'],cmap='Blues',bins=(15,15))
_ = plt.colorbar()
_ = plt.ylabel('Plastron')

_ = plt.suptitle('Native Males Histogram ')
_ = plt.subplot(2,2,3)
#_ = plt.scatter(males['Annuli'], males['Weight'],s=10)
_ = plt.hist2d(males['Annuli'], males['Weight'],cmap='Blues',bins=(15,15))
_ = plt.colorbar()
_ = plt.xlabel('Annuli')
_ = plt.ylabel('Weight')
_ = plt.tight_layout(rect=[0, 0.03, 1, 0.95])

Plotting Histogram Weight, Carapace and Plastron with 336 samples...

Weight distributions across 2 separate populations

The hypothesis: same distribution in all capture locations proved true

#Hyposis: same distribution in all capture locations
native_portland = natives[(natives['Capture Location']=='Mason Flats')]
native_gresham = natives[(natives['Capture Location']=='Gresham') ]
#&(natives['Gender']=='f')
genders = ['f','m']
dimension = 'Weight'
_ = plt.figure(6)
i = 0
for gender in genders:
    i += 1
    _ = plt.subplot(1,2,i)
    for _ in range(100):
        # Generate permutation samples
        native_portland_gender = native_portland[native_portland['Gender']==gender]
        native_gresham_gender = native_portland[native_portland['Gender']==gender]
        perm_sample_1, perm_sample_2 = hlp.permutation_sample(native_portland_gender[dimension],native_gresham_gender[dimension])

        # Compute ECDFs
        x_1, y_1 = hlp.ecdf(perm_sample_1)
        x_2, y_2 = hlp.ecdf(perm_sample_2)

        # Plot ECDFs of permutation sample
        _ = plt.plot(x_1, y_1, marker='.', linestyle='none',
                     color='green', alpha=0.02)
        _ = plt.plot(x_2, y_2, marker='.', linestyle='none',
                     color='orange', alpha=0.02)

    # Create and plot ECDFs from original data
    x_1, y_1 = hlp.ecdf(native_portland_gender[dimension])
    x_2, y_2 = hlp.ecdf(native_portland_gender[dimension])
    _ = plt.plot(x_1, y_1, marker='.', linestyle='none', color='green',label='Portland')
    _ = plt.plot(x_2, y_2, marker='.', linestyle='none', color='orange',label='Gresham')
    _ = plt.legend()
    _ = plt.title(gender)
    # Label axes, set margin, and show plot
    plt.margins(0.02)
    _ = plt.xlabel(dimension)
    _ = plt.ylabel('ECDF')
plt.show()

#Hyposis: DIFFERENT distribution between genders
native_females = natives[(natives['Gender']=='f')]
native_males = natives[(natives['Gender']=='m')]
#&(natives['Gender']=='f')
dimension = 'Weight'
_ = plt.figure(7)
for _ in range(100):
    # Generate permutation samples
    perm_sample_1, perm_sample_2 = hlp.permutation_sample(native_females[dimension],native_males[dimension])

    # Compute ECDFs
    x_1, y_1 = hlp.ecdf(perm_sample_1)
    x_2, y_2 = hlp.ecdf(perm_sample_2)

    # Plot ECDFs of permutation sample
    _ = plt.plot(x_1, y_1, marker='.', linestyle='none',
                 color='red', alpha=0.02)
    _ = plt.plot(x_2, y_2, marker='.', linestyle='none',
                 color='blue', alpha=0.02)

# Create and plot ECDFs from original data
x_1, y_1 = hlp.ecdf(native_females[dimension])
x_2, y_2 = hlp.ecdf(native_males[dimension])
_ = plt.plot(x_1, y_1, marker='.', linestyle='none', color='red',label='female')
_ = plt.plot(x_2, y_2, marker='.', linestyle='none', color='blue',label='male')
_ = plt.legend()
_ = plt.title('Permutation Sampling by Gender')
# Label axes, set margin, and show plot
plt.margins(0.02)
_ = plt.xlabel(dimension)
_ = plt.ylabel('ECDF')
plt.show()

Correlations between annuli, size and weight.

Annuli is less of a predictor of an animals size/weight, however, size and weight are strongly correlated.

corrleations = native_males.loc[slice(None),['Annuli','Weight','Carapace','Plastron']].corr()

_ = sns.heatmap(corrleations,annot=True)
_ = plt.xticks(rotation=45)
_ = plt.show()

Distribution of weights and sizes by sex, separating out gravid and non-gravid females

.

#
colors = {
    'f_spr': sns.xkcd_rgb['soft pink'],
    'f_fal': sns.xkcd_rgb['hot pink'],
    'f_gra': sns.xkcd_rgb['gold'],
    'm_spr': sns.xkcd_rgb['baby blue'],
    'm_fal': sns.xkcd_rgb['denim blue'],
    'unknown': sns.xkcd_rgb['very light green'],
    'f': sns.xkcd_rgb['hot pink'],
    'm': sns.xkcd_rgb['denim blue'],
}
natives_melted13 = natives.copy()
natives_melted13['Weight / 10'] = natives_melted13.Weight.div(10)
natives_melted13 = pd.melt(natives_melted13[['Carapace','Plastron','Weight / 10','gender_plus']],'gender_plus',var_name='measurement')

sns.swarmplot(x="measurement", y="value", hue="gender_plus", data=natives_melted13,palette=colors,size=3)


someColumns = [
    'Carapace',
    'Plastron',
    'Weight',
    'gender_plus'
]
stats_df = natives[someColumns].groupby('gender_plus').mean()

gender_stats_parms = {
    'cellText': stats_df.values,
    'rowLabels': stats_df.index.values,
    'colLabels': ["Carapace (Mean)","Plastron (Mean)","Weight (Mean)"],
    'loc': 'top',}
    
gender_stats = plt.table(**gender_stats_parms)
plt.xticks([])
plt.tight_layout(True)

_ = plt.show()

Scatterplot graph showing the correlation of annuli with size/weights of animals by sex with gravid females separated from non-gravid females.

Same as above - showing the relationship between size/weight and age for gravid females, non-gravid females, males, and unknowns. Can we include the averages on this graph too?

display("Mean values per gender and location")
someColumns = [
    'Carapace',
    'Plastron',
    'Weight',
    'gender_plus',
    'Capture Location',
]
stats_df = natives[natives['Gender']!='unknown'][someColumns].groupby(['gender_plus','Capture Location']).mean()
display(stats_df)

'Mean values per gender and location'

		Carapace	Plastron	Weight
gender_plus	Capture Location
f	Gresham	157.554901	150.427460	641.449280
	Mason Flats	161.417999	153.246506	636.599976
	Whitaker Ponds	200.080002	190.047501	963.512512
f_gra	Gresham	195.779999	188.431595	1088.239990
	Mason Flats	202.638748	192.207504	1125.375000
	Whitaker Ponds	219.175003	209.202499	1343.500000
m	Gresham	125.340240	117.063057	285.565979
	Mason Flats	145.819061	135.210007	408.566040
	Whitaker Ponds	182.255005	169.107498	721.000000

# scatterplot with linear digression 
#sns.set()
# import statsmodels
natives14 = natives[natives['Gender']!='unknown']
is_robust = False


### Swarmplot with Hue and Linear Digression 
natives14 = natives14[natives14['Annuli']!=0]
#colors = [sns.xkcd_rgb['pale red'],sns.xkcd_rgb['denim blue'],sns.xkcd_rgb['medium green']]
plt.rcParams['figure.dpi'] = 500
print ("NEW ANNULI - Plotting scatter plot with hue  " + str(natives14.Weight.count()) + ' samples...')
lmplotParams = {
    'x': 'Annuli', 
    'y': 'Carapace', 
    'hue': 'gender_plus',
    'col': "Capture Location",
    'data': natives14,
    'palette': colors,
    'scatter': True,
    'fit_reg': False,
    'lowess': True,
    'legend_out': False,
    'x_jitter': .5,
    'y_jitter': .1,
    'scatter_kws': {"s": 7,"alpha": .7},
}
sns.set()
#lowess=True, too bus
_ = sns.lmplot(**lmplotParams)
_ = plt.ylabel('Carapace')
lmplotParams['y'] = 'Plastron'
_ = sns.lmplot(**lmplotParams)
_ = plt.ylabel('Plastron')
lmplotParams['y'] = 'Weight'
_ = sns.lmplot(**lmplotParams)
_ = plt.ylabel('Weight')



plt.tight_layout(True)

NEW ANNULI - Plotting scatter plot with hue  774 samples...

Scatterplot showing the distribution of sizes and weights as a function of which season the female data was collected.

Same as the two graphs above. Shows distribution of sizes and weights and separates females into three groups (non-gravid spring, gravid spring, and fall measurements) and males and unknowns.

natives_melted15 = natives14.copy()
natives_melted15['Weight / 10'] = natives_melted15.Weight.div(10)
natives_melted15 = pd.melt(natives_melted15[['Carapace','Plastron','Weight / 10','gender_seasons']],'gender_seasons',var_name='measurement')

_ = sns.swarmplot(x="measurement", y="value", hue="gender_seasons", data=natives_melted15,palette=colors,size=3)
_ = plt.show()

Comparison of size/weight distributions across seasons between Gresham and Mason Flats populations

Comparison of size/weight distributions by sex for animals caught in the spring vs. fall between two populations (Gresham and Mason Flats).

# import statsmodels
natives16 = natives14[natives14['Gender']!='unknown']
is_robust = False


### Swarmplot with Hue and Linear Digression 
natives16 = natives16[natives16['Annuli']!=0]
#colors = [sns.xkcd_rgb['pale red'],sns.xkcd_rgb['denim blue'],sns.xkcd_rgb['medium green']]
plt.rcParams['figure.dpi'] = 500
print ("NEW ANNULI - Plotting scatter plot with hue  " + str(natives16.Weight.count()) + ' samples...')
lmplotParams = {
    'x': 'Annuli', 
    'y': 'Carapace', 
    'hue': 'gender_seasons',
    'col': "Capture Location",
    'data': natives16,
    'palette': colors,
    'scatter': True,
    'fit_reg': False,
    #'lowess': True,
    'legend_out': False,
    'x_jitter': .5,
    'y_jitter': .1,
    'scatter_kws': {"s": 7,"alpha": .7},
}
_ = sns.lmplot(**lmplotParams)
_ = plt.ylabel('Carapace')
lmplotParams['y'] = 'Plastron'
_ = sns.lmplot(**lmplotParams)
_ = plt.ylabel('Plastron')
lmplotParams['y'] = 'Weight'
_ = sns.lmplot(**lmplotParams)
_ = plt.ylabel('Weight')

NEW ANNULI - Plotting scatter plot with hue  774 samples...

Comparison of size/weight distributions across seasons between Gresham and Mason Flats populations

Same as above graphs, but with higher-order polynomial regression lines. Comparison of size/weight distributions by sex for animals caught in the spring vs. fall between two populations (Gresham and Mason Flats). Males show no real difference in distribution between populations or seasons. Females have a greater diversity of sizes/weights and are heavier in fall vs. spring, with gravid females being the heaviest on average.

#lmplotParams['lowess'] = False
lmplotParams['order'] = 2
lmplotParams['ci'] = None
lmplotParams['fit_reg'] = True
lmplotParams['truncate'] = True

lmplotParams['y'] = 'Carapace'
_ = sns.lmplot(**lmplotParams)
_ = plt.ylabel('Carapace')
plt.show()
lmplotParams['y'] = 'Plastron'
_ = sns.lmplot(**lmplotParams)
_ = plt.ylabel('Plastron')
plt.show()
lmplotParams['y'] = 'Weight'
_ = sns.lmplot(**lmplotParams)
_ = plt.ylabel('Weight')
plt.show()

/Users/kkopowski/.asdf/installs/python/3.8.12/lib/python3.8/site-packages/seaborn/regression.py:256: RankWarning: Polyfit may be poorly conditioned
  yhat = reg_func(x, y)
/Users/kkopowski/.asdf/installs/python/3.8.12/lib/python3.8/site-packages/seaborn/regression.py:256: RankWarning: Polyfit may be poorly conditioned
  yhat = reg_func(x, y)

/Users/kkopowski/.asdf/installs/python/3.8.12/lib/python3.8/site-packages/seaborn/regression.py:256: RankWarning: Polyfit may be poorly conditioned
  yhat = reg_func(x, y)
/Users/kkopowski/.asdf/installs/python/3.8.12/lib/python3.8/site-packages/seaborn/regression.py:256: RankWarning: Polyfit may be poorly conditioned
  yhat = reg_func(x, y)

/Users/kkopowski/.asdf/installs/python/3.8.12/lib/python3.8/site-packages/seaborn/regression.py:256: RankWarning: Polyfit may be poorly conditioned
  yhat = reg_func(x, y)
/Users/kkopowski/.asdf/installs/python/3.8.12/lib/python3.8/site-packages/seaborn/regression.py:256: RankWarning: Polyfit may be poorly conditioned
  yhat = reg_func(x, y)